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Modeling Interactions Between Electrical Activity and Second-Messenger Cascades in Aplysia Neuron R15

Center for Computational Biomedicine, Department of Neurobiology and Anatomy, The University of Texas–Houston Medical School, Houston, Texas 77030

Submitted 13 August 2003; accepted in final form 19 December 2003

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

 
The biophysical properties of neuron R15 in Aplysia endow it with the ability to express multiple modes of oscillatory electrical activity, such as beating and bursting. Previous modeling studies examined the ways in which membrane conductances contribute to the electrical activity of R15 and the ways in which extrinsic modulatory inputs alter the membrane conductances by biochemical cascades and influence the electrical activity. The goals of the present study were to examine the ways in which electrical activity influences the biochemical cascades and what dynamical properties emerge from the ongoing interactions between electrical activity and these cascades. The model proposed by Butera et al. in 1995 was extended to include equations for the binding of Ca²⁺ to calmodulin (CaM) and the actions of Ca²⁺/CaM on both adenylyl cyclase and phosphodiesterase. Simulations indicated that levels of cAMP oscillated during bursting and that these oscillations were approximately antiphasic to the oscillations of Ca²⁺. In the presence of cAMP oscillations, brief perturbations could switch the electrical activity between bursting and beating (bistability). Compared with a constant-cAMP model, oscillations of cAMP substantially expanded the range of bistability. Moreover, the integrated electrical/biochemical model simulated some early experimental results such as activity-dependent inactivation of the anomalous rectifier. The results of the present study suggest that the endogenous activity of R15 depends, in part, on interactions between electrical activity and biochemical cascades.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

 
The bursting neuron R15 of Aplysia provides an excellent model system for both empirical and computational studies of neuronal dynamics and their regulation (Baxter et al. 2000). R15 generates robust endogenous bursting activity. In addition, the cell can generate continuous spiking known as beating. The expression of different modes of electrical activity can be controlled, in part, by applying stimuli that hyperpolarize or depolarize the membrane potential and/or by modulatory transmitters that activate intracellular biochemical cascades (Carpenter et al. 1978; Drummond et al. 1980; Gospe and Wilson 1980; Levitan et al. 1987; Lotshaw et al. 1986; Mathieu and Roberge 1971). The electrical and biochemical properties of R15 have been studied in great detail. Various ionic conductances have been identified and characterized and their role in rhythmicity in R15 has been investigated (Adams 1985; Adams and Benson 1985; Adams and Gage 1979). In addition, biochemical cascades that modulate membrane conductances have been examined and quantified (Kramer et al. 1988; Lotshaw and Levitan 1988).

One method that has proven advantageous for analyzing this wealth of empirical data is the development of detailed computational models of R15 (Adams and Benson 1985; Bertram 1994; Butera et al. 1995; Canavier et al. 1991; Chay 1983; Rinzel and Lee 1987). These models focused primarily on membrane conductances and the dynamic electrical properties of R15 but to a lesser extent include descriptions of the biochemical processes in the cell. Although these models have provided important insights into how bursts are generated and modulated, they do not include direct interactions between electrical activity and biochemical processes. It has become increasingly clear that the electrical activity of a neuron is not isolated from intracellular biochemical processes. The electrical activity primarily through Ca²⁺ influx modulates second-messenger cascades, and in turn, the second-messenger cascades influence the electrical activity by modulating ionic currents. For example, 2 enzymes that catalyze cAMP synthesis and degradation, adenylyl cyclase (AC) and phosphodiesterase (PDE), are subject to modulation by Ca²⁺-bound calmodulin (Ca²⁺/CaM) and free Ca²⁺ (Abrams et al. 1991; Kramer et al. 1988; Levitan and Levitan 1988). On one hand, Ca²⁺ influx during bursting in R15 causes an increase of intracellular concentration of free Ca²⁺ (Gorman and Thomas 1978), which presumably changes the activity of AC and PDE, and thereby influences the concentration of cAMP ([cAMP]). On the other hand, cAMP presumably influences the intracellular concentration of Ca²⁺ ([Ca²⁺]_i) by modulating Ca²⁺ conducting ionic currents such as the slow inward current (I_SI) (Lotshaw and Levitan 1988). By I_SI, cAMP may also significantly modulate the electrical activity of R15 because I_SI is believed to play a key role in generating the bursting activity (Adams and Levitan 1985; Canavier et al. 1991; Gorman et al. 1982).

However, the ways in which such interactions between electrical and biochemical processes quantitatively contribute to neuronal function are not well understood. The model developed by Butera et al. (1995) implemented a mechanism for cAMP modulation of 2 ionic currents, the anomalous rectifier (I_R) and I_SI. The model was successful in explaining how serotonin (5-HT) and dopamine (DA) affect the endogenous activity of R15. However, the interactions implemented in Butera et al.'s model were only unidirectional, from the second-messenger cascades to the electrical activity. Influences of the electrical activity on the second-messenger cascades were not considered.

Given the potential interactions between bursting and the levels of cAMP (Kramer et al. 1988), it is necessary to extend the model by Butera et al. (1995) to include descriptions of Ca²⁺-dependent modulation of cAMP. The current study describes such an integrated electrical/biochemical (IE/B) model that includes detailed descriptions of interactions between Ca²⁺/CaM and the cAMP regulation. With these modifications, we found that the cAMP levels oscillated during bursting, indicating that the second-messenger cascades were effectively influenced by the electrical activity. In addition, cAMP oscillations contributed to the dynamic electrical properties of the neuron.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

 
The IE/B model was based on the model of Butera et al. (1995), which evolved from the model of Canavier et al. (1991). The IE/B model consists of 2 parts: a Hodgkin–Huxley type electrical circuit that describes membrane dynamics and a fluid compartment model that describes biochemical processes associated with the regulation of Ca²⁺ and cAMP. No changes were made to the previous descriptions of the electrical circuit. All equations and parameters in the electrical circuit were identical to those of Butera et al. (1995). The parameters in the model of Butera et al. were either adopted from published literature or obtained by curve fitting empirical data (mostly from R15 or other Aplysia neurons). Some of the rate constants, such as that in I_Na and I_K, were scaled to account for the difference between the temperature at which the model operates (22°C) and the temperatures in some of the experiments (13.5–16°C). Only in a few cases, parameters were adjusted by modest degrees to make the model more stable or behave more consistently with empirical results.

In the present study, the major changes in this model were made to the descriptions of Ca²⁺ and cAMP regulation and are described in detail (see below). Only I_SI and I_R (the 2 currents modulated by cAMP) are described below. All other currents and features of the model are presented in the APPENDIX.

Regulation of calcium

The description of the regulation of Ca²⁺ was extended to include CaM, which mediates some of the effects of Ca²⁺ on AC and PDE (Fig. 1). CaM also serves as a Ca²⁺ buffer in addition to the first-order Ca²⁺ buffer implemented in the previous models. The first-order Ca²⁺ buffer remained in the IE/B model and represents an array of cytosolic Ca²⁺ buffers, such as endoplasmic reticulum and mitochondria. Intracellular Ca²⁺ concentration was regulated by Ca²⁺ influx by Ca²⁺ conducting ionic channels, pumps, and uptake/release of Ca²⁺ to/from the Ca²⁺ buffers. The material balance equation for [Ca²⁺]_i was

(1)

where the overdot of a variable means its time derivative, I_NaCa is the Na⁺–Ca²⁺ exchanger, I_SI is the slow inward current, I_Ca is the fast Ca²⁺ current, I_CaP is the Ca²⁺ pump, I_NS is the nonspecific current, V is the membrane potential, E_Ca is the reversal potential of Ca²⁺, E_NS is the reversal potential for the nonspecific current, 0.197(V - E_Ca)/(V - E_NS) is the fraction of I_NS conducting Ca²⁺ {0.197(V - E_Ca)/(V - E_NS) was used because the multiplier for I _NS in Ca²⁺ balance is given by [g_NS,Ca(V - E_Ca)]/[g_NS(V - E_NS)] and the value for g_NS,Ca/g_NS is 0.197}, 2Vol_iF is used here because Ca²⁺ is a divalent cation and carries 2 positive charges, n_B is the number of binding sites per molecule on the first-order Ca²⁺ buffer, [B]_i is the concentration of the first-order Ca²⁺ buffer, O_C is the fraction of binding sites occupied by Ca²⁺, and [Ca]_CaM is the total concentration of Ca²⁺ buffered by CaM (see below).

View larger version (30K): [in this window] [in a new window]   FIG. 1. Interaction between fluid components and membrane conductances. In the integrated electrical/biochemical (IE/B) model, the change in intracellular Ca2+ concentration was determined by: 1) the Ca2+ flux by Ca2+ components of various ion channels and pumps/exchangers, and 2) uptake/release of Ca2+ by Ca2+ buffers, which includes both CaM and a first-order Ca2+ buffer. Synthesis and degradation of cAMP were catalyzed by adenylyl cyclase (AC) and phosphodiesterase (PDE), respectively. AC was activated by serotonin (5-HT) and Ca2+/CaM and inhibited by free Ca2+ (only at high Ca2+ concentration), whereas PDE activity was enhanced by Ca2+/CaM. Two currents that play a key role in rhythmicity, IR and ISI, are upregulated by cAMP. Dashed lines indicate pathways/components added in the IE/B model.

(2)

(3)

(4)

(5)

(6)

(7)

where k_1f–k_4f are the forward rate constants and k_1b–k_4b are the backward rate constants in the 4 steps of Ca²⁺ binding to CaM. Their values are given in the APPENDIX.

CaM needs to bind at least 3 Ca²⁺ to be active (Blumenthal and Stull 1980; Rasmussen and Barrett 1984). Therefore, among the 4 species of Ca²⁺-bound CaM, only Ca₃CaM and Ca₄CaM were effective in regulating the AC and PDE activities. For simplicity, Ca₃CaM and Ca₄CaM were considered to be equally efficient in regulating the AC and PDE activities. However, because [Ca₃CaM] was approximately 3- to 4-fold higher than [Ca₄CaM], Ca₃CaM played the major role in regulating cAMP synthesis and degradation.

In all simulations except for Fig. 5, the total concentration of CaM was set to 11.25 µM. This total concentration of CaM was in agreement with empirical data obtained from mammalian systems (Chafouleas et al. 1982; Tansey et al. 1994). To keep the total concentration of Ca²⁺ buffer the same as that in the previous models, the concentration of the first-order Ca²⁺ buffer was reduced from 112.5 to 101.25 µM. Thus, 10% of the total Ca²⁺ buffer was CaM and the remaining 90% was the first-order Ca²⁺ buffer implemented by the previous models (Butera et al. 1995; Canavier 1991).

View larger version (30K): [in this window] [in a new window]   FIG. 5. Effect of CaM composition within the Ca2+ buffering system. Percentage of total CaM within the whole Ca2+ buffering system was varied from 2.8 µM (2.5% CaM) to 56 µM (50% CaM). Bursting activity and oscillations of [Ca2+] did not change much with such variations. However, oscillations of [cAMP] were quite different at different CaM levels. Both the amplitude of [cAMP] oscillations and the average cAMP concentration decreased when the total CaM was increased. For Ca3CaM and Ca4CaM, both their amplitude of oscillations and their average concentration proportionally increased when the CaM level was increased.

Synthesis and degradation of cAMP were catalyzed by AC and PDE, respectively

(8)

where v_AC is AC activity and v_PDE is PDE activity.

Although the Ca²⁺-dependent regulation of AC has not been specifically examined in R15, the regulation has been examined in the central nervous system of Aplysia (Abrams et al. 1991; Eliot et al. 1989; Yovell et al. 1992). AC activity is believed to be upregulated by Ca²⁺-bound CaM at low [Ca²⁺] (<10 µM) and downregulated by free Ca²⁺ at high [Ca²⁺] (>10 µM; Abrams et al. 1991). This bimodal regulatory mechanism was simulated by the product of 2 first-order Michaelis–Menton-type kinetic equations, one fitting the upregulation half of the bimodal curve and the other fitting the downregulation half of the bimodal curve (Fig. 2A). In addition, the AC activity was also modulated by 5-HT, which is described by a first-order kinetic equation. Overall, the function of AC is

(9)

where _AC is basal level AC activity (0.6 nM/ms), is a constant (0.5), K_mod,Ca is a scaling factor (1.0) that defines the dynamic range of AC activity when activated by Ca²⁺, K_P,CaM is the half-maximal value of [Ca₃CaM] + [Ca₄CaM] for the upregulation of AC by Ca²⁺/CaM (0.968 nM; derived from Abrams et al. 1991), K_N,Ca is the half-maximal value of [Ca²⁺] for the downregulation of AC by free Ca²⁺ (75 µM; derived from Abrams et al. 1991), K_mod,5HT is a scaling factor (1.2) that defines the dynamic range of AC activity when activated by 5-HT, and K_5HT is the half-maximal value of [5-HT] (6 µM; Butera et al. 1995). Values for , K_mod,Ca, K_N,Ca, and K_P,CaM were derived by fitting the available data of AC.

View larger version (17K): [in this window] [in a new window]   FIG. 2. Parameterizing the IE/B model. Filled circles represent empirical data and the curves are the fitting functions that were used in the model. A: regulation of AC by Ca2+ and Ca2+/CaM. Upregulation of AC at low Ca2+ concentration (<10 µM) was mediated by Ca2+ bound CaM, whereas down-regulation of AC at high Ca2+ concentration (>10 µM) was mediated by free Ca2+. Experimental data were fitted by the product of 2 first-order Michaelis–Menton-type kinetic equations. B: regulation of PDE by Ca2+. Experimental data were fitted by a first-order Michaelis–Menton-type kinetic equation (see text and APPENDIX).

(10)

where _PDE is basal level PDE activity (2.4 nM/ms), is a constant (0.4), K_mod,PDE is a scaling factor (1.2) that defines the dynamic range of PDE activity when activated by Ca²⁺, and K_D is the half maximal value of [Ca₃CaM] + [Ca₄CaM] (0.968 nM; derived from Kramer et al. 1988). Considering the rate of this enzyme reaction is also determined by the concentration of its substrate, cAMP, the function of PDE activity includes terms for the [cAMP], where K_PDE is the half maximal value of [cAMP] (3.0 µM; derived from Kramer et al. 1988). (Similarly, the function of AC activity should also be dependent on where v the concentration of its substrate, [ATP]. However, we assume that [ATP] is relatively constant, given the abundance of this common molecule. Therefore the function of AC activity does not include the term of [ATP].)

Although both AC and PDE are upregulated by Ca²⁺/CaM when [Ca²⁺] is low, the sensitivity of these 2 enzymes to Ca²⁺ is quite different. Between 0.2 and 0.5 µM, which is the normal operating range of [Ca²⁺] during bursting (Gorman et al. 1981), the dose–response curve of PDE is much steeper than the dose–response curve of AC. For example, the PDE activity increases almost 300% from [Ca²⁺] = 0.1 µM to [Ca²⁺] = 1.0 µM, whereas the AC activity concomitantly increases only about 100%. This difference in sensitivity is of great importance to oscillations of [cAMP] (see RESULTS).

Currents modulated by cAMP

SLOW INWARD CA<SUP>2+SUP> CURRENT (I_SI). The I_SI is the key current responsible for generating bursting in the model. It is voltage activated and Ca²⁺ inactivated. In the present model as well as in the previous models (Butera et al. 1995; Canavier et al. 1991), bursting begins with the voltage activation of I_SI, continues as Ca²⁺ accumulates internally, and ends with the Ca²⁺-dependent inactivation of I_SI. I_SI is modulated by cAMP and dopamine (DA) (Lotshaw and Levitan 1988). As described previously by Butera et al. (1995), I_SI is described by

where _SI is the maximal conductance for this current at basal concentrations of cAMP and DA (0.65 µS), K_SI,Ca is the half-inactivation value of [Ca²⁺] (0.025 µM), s is a voltage-dependent gating variable (dimensionless), F_SI,mod is a dimensionless modulation term that represents the effects of cAMP and dopamine on I_SI, and K_DA is the half-inactivation value for DA on I_SI (0.2 mM). The concentration of DA was set to 0 in the present study. K_SI,mod (5.5), K_SI,cAMP (4.2 µM), and D_SI,cAMP (0.35 µM) are parameters for Boltzmann-type characterization of the binding site for cAMP. They determine modulation of I_SI by cAMP.

ANOMALOUS-RECTIFIER POTASSIUM CURRENT (I_R). This current is the dominant current at very hyperpolarization potentials (< -70 mV), which occur between bursts. The I_R conductance is modulated by [cAMP], the action of which is believed to recruit additional ion channels (Gunning 1987). As described previously by Butera et al. (1995), I_R is described by

where _R is the maximal unmodulated conductance (0.18 µS) and F_R,mod is a dimensionless modulation term that represents the effect of cAMP on I_R. Similar to I_SI, K_R,mod (1.5), K_R,cAMP (1 µM), and D_R,cAMP (0.4 µM) are constants that determine modulation of I_R by cAMP.

All parameter values in the above equations (currents modulated by cAMP) are identical to those in Butera et al. (1995; see also Canavier et al. 1991).

Computational methods

Simulations were run on a PC-type computer with a Microsoft Windows operating system and the dynamic simulation tool XPP (Ermentrout 2002). The integration algorithm is Cvode with a tolerance of 10^-7 and a minimum time step of 10^-12 s.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

 
Baseline activity

The IE/B model was first examined to determine whether it generated baseline activity similar to that demonstrated in previous studies (Butera et al. 1995; Canavier et al. 1991). As illustrated in Fig. 3, the bursting activity generated by the IE/B model faithfully replicated previously simulated activity (Butera et al. 1995; Canavier et al. 1991), which was similar to experimental results (e.g., Arvanitaki and Charlazonitis 1968). The average cycle period (17–18 s), the amplitude of spikes (50–55 mV), and the number of spikes per burst (17) were all in agreement with the experiments and the previous models. However, there were some subtle differences in the bursts generated by the IE/B model and the model of Butera et al. (see figure legends). The 5-HT response was also similar to the previous models (simulation not shown). The IE/B model also displayed the "parabolic" bursting pattern (the spike frequency first increased then decreased during a burst) characteristic of R15.

View larger version (25K): [in this window] [in a new window]   FIG. 3. Comparison between the IE/B model (A) and Butera et al. (1995) model (B). Bursting activity and the oscillations of the Ca2+ levels generated by the 2 models were very similar, indicating that the extension to the model did not change the baseline activity. In the IE/B model, the concentration of cAMP ([cAMP]) oscillated between 0.95 and 1.05 µM and the oscillations were approximately antiphasic to the oscillations of the concentration of intracellular free Ca2+ ([Ca2+]). In contrast, [cAMP] did not change in the Butera et al. (1995) model because it was described by a constant rather than a dynamic variable (1.0 µM). In the IE/B model, concentrations of Ca3CaM ([Ca3CaM]) and Ca4CaM ([Ca4CaM]) were introduced as 2 new variables to mediate the Ca2+ regulation of the activities of AC and PDE. [Ca3CaM] and [Ca4CaM] also oscillated with the bursting activity in phase with free Ca2+. C: bursts outlined by the box in A and B are illustrated in greater detail here. There were subtle differences in the bursts generated by the IE/B model (C1) and the model of Butera et al. (C2). For example, the IE/B model produced one additional spike (*) during a control burst.

Parallel to oscillations of [Ca²⁺], levels of Ca²⁺/CaM also oscillated with the bursting activity and had the same phase as the free Ca²⁺. [Ca₃CaM] oscillated between 0.05 and 0.20 µM and [Ca₄CaM] oscillated between 0.01 and 0.07 µM. The relative amplitudes of the oscillations of [Ca₃CaM] and [Ca₄CaM] were greater than the oscillations of [Ca²⁺]. Because Ca₃CaM and Ca₄CaM mediated the effect of Ca²⁺ on AC and PDE, this implied that such effects were amplified at the level of CaM.

In the model of Butera et al. (1995), AC and PDE were not subject to regulation by Ca²⁺ or Ca²⁺/CaM. Thus, their activities were constant during bursting. In the IE/B model, both AC and PDE were regulated by Ca²⁺/CaM and the enzyme activities of AC and PDE were simulated by Ca²⁺/CaM-dependent functions (see METHODS). Consequently, the activities of AC and PDE became dynamic during bursting and this in turn influenced the dynamics of cAMP (see below).

Levels of cAMP oscillate during bursting

Given that the IE/B model faithfully reproduced previous empirical and computational results, we next examined the ways in which the bursting activity influenced the behavior of cAMP. Levels of cAMP oscillated during bursting, with a range of between 0.95 and 1.05 µM and an average of 1.0 µM. Although oscillations of [cAMP] have not been detected empirically in R15, it is likely they exist given that synthesis and degradation of cAMP are regulated by Ca²⁺/CaM and levels of Ca²⁺/CaM are oscillating during bursting. Oscillations of [Ca²⁺/CaM] would inevitably lead to the oscillations of [cAMP].

Interestingly, however, oscillations of [cAMP] were almost antiphasic to oscillations of [Ca²⁺] or [Ca²⁺/CaM]. The peaks of the cAMP oscillations occurred at the troughs of the Ca²⁺ oscillations. Because oscillations of [Ca²⁺/CaM] drove the oscillations in [cAMP], this antiphasic relationship indicates that the net effect of Ca²⁺/CaM must be to provide a negative drive to the levels of cAMP (see below).

Relative contributions of AC and PDE to oscillations of levels of cAMP

To understand the ways in which oscillations of [Ca²⁺/CaM] drive oscillations of [cAMP], it is necessary to elucidate the roles and relative contributions of AC and PDE to oscillations of [cAMP]. AC and PDE have opposite effects on the levels of cAMP and the relative contributions of these 2 enzymes determine the ways in which the levels of cAMP oscillate.

In Fig. 4A, activities of AC (orange) and PDE (blue) were plotted. They both oscillated in phase with Ca²⁺. However, because PDE was more sensitive to [Ca²⁺/CaM] than AC (Fig. 2), the PDE activity oscillated at a slightly greater amplitude than the AC activity. Although the difference was small, it was enough to cause degradation of cAMP to outpace its synthesis during a burst when the Ca²⁺ level was high. In contrast, synthesis of cAMP outpaced its degradation between 2 bursts when the Ca²⁺ level was low. Consequently, the cAMP levels decreased during bursts and increased between the bursts. During normal baseline bursting activity, PDE activity appears to be the dominant mechanism at causing the cAMP oscillations.

View larger version (38K): [in this window] [in a new window]   FIG. 4. Contribution of AC and PDE to oscillations of [cAMP]. A: in control condition, both the AC activity and PDE activity oscillated in phase with free Ca2+. However, because the PDE activity (blue line) was more sensitive to Ca2+/CaM than the AC activity (orange line), the amplitude of the oscillations of the PDE activity was slightly larger than the amplitude of the oscillations of the AC activity. This difference between the 2 enzymes determined the phase and the amplitude of the oscillations of [cAMP]. B: if the Ca2+/CaM regulation of PDE was eliminated, leaving AC the only enzyme regulated by Ca2+/CaM, oscillations of the AC activity became larger in amplitude than oscillations of the PDE activity. As the result, both the amplitude and the phase of the oscillations of [cAMP] were altered. Note the oscillation of the PDE activity here was driven by the oscillations of [cAMP]. C: if Ca2+/CaM regulation of AC was eliminated, leaving PDE the only enzyme regulated by Ca2+/CaM, oscillations of [cAMP] were antiphasic to oscillations of [Ca2+], although the amplitude of these oscillations was much larger compared with A. Bursting activity, most notably the cycle period, was greatly influenced in this situation. Note the AC activity did not oscillate here because it was [cAMP] independent.

When Ca²⁺/CaM regulation of AC was eliminated (Fig. 4C), cAMP oscillations were in phase with PDE activity. Here the cAMP oscillation was driven by PDE activity only because the AC activity no longer oscillated. The amplitude of the [cAMP] oscillations was large because of the constant AC activity.

From these simulations we demonstrated that the relative strength of oscillations in the activities of AC and PDE defined both the phase and amplitude of oscillations of [cAMP]. Empirically, this result suggests that eliminating or changing Ca²⁺/CaM regulation of AC or PDE could lead to change of the phase and/or the amplitude of cAMP oscillations.

CaM composition within Ca²⁺ buffering system

In the above simulations, the concentration of total CaM, which included the 4 species of Ca²⁺ bound CaM and unbound CaM, was 11.25 µM, which was 10% of the concentration of the whole Ca²⁺ buffering system. The remaining 90% was the first-order Ca²⁺ buffer of Butera et al. (1995). This concentration of total CaM was close to that found empirically in other systems (Chafouleas et al. 1982; Tansey et al. 1994) and the concentration at which activities of AC and PDE were assayed in Aplysia neurons (Yovell et al. 1992). However, the exact concentration of total CaM in R15 is still unknown. Moreover, the local concentration of CaM can vary greatly because of the compartmentalization within the cell (Luby-Phelps et al. 1995). The following simulations were designed to examine the ways in which the model responded to variations in the CaM level by varying the percentage of CaM in the whole Ca²⁺ buffering system from the original 10% (11.2 µM) down to 2.5% (2.8 µM) and up to 50% (56 µM).

There were some modest changes in the bursting of membrane potential when the total CaM was varied (Fig. 5). The most obvious effects were broadening of the bursts and increase in the number of spikes per burst that occurred when the total CaM was increased. There was also some slight increase in the cycle period that accompanied the increase in total CaM. Overall, varying the total CaM did not substantially alter the bursting activity. Moreover, even when the first-order Ca²⁺ buffering system was completely replaced by CaM (100% CaM), the model continued to generate bursting activity (simulation not shown).

Oscillations of [Ca²⁺] also did not change much except that the average Ca²⁺ concentration increased slightly from 0.32 to 0.35 µM. For cAMP, however, both the amplitude of its oscillations and the average concentration were substantially altered. The amplitude of oscillations varied from more than 0.1 µM (2.5 and 5% CaM) to less than 0.05 µM (50% CaM), whereas the average [cAMP] varied from 1.13 µM (2.5% CaM) to 0.93 µM (50% CaM). At high total CaM, this decrease in average [cAMP] was caused by high [Ca₃CaM] and [Ca₄CaM].

For [Ca₃CaM] and [Ca₄CaM], both the amplitude of oscillations and the average concentration were proportionally increased when the total CaM was increased. However, larger oscillations in [Ca₃CaM] and [Ca₄CaM] at higher Ca²⁺/CaM levels correspond to smaller oscillations in [cAMP]. This was so because at higher Ca²⁺/CaM levels, the dose–response curve of the PDE activity (Fig. 2B) became much less steep. The overall effect was that the difference between the AC activity and the PDE activity became so small that even larger oscillations in [Ca₃CaM] and [Ca₄CaM] would not lead to larger oscillations in [cAMP].

cAMP levels are lower during bursting than during the silent state

Previous experimental results indicated that cAMP levels in R15 were lower during bursting than during the silent state (Kramer et al. 1988). This result was simulated here by switching the model between the bursting mode and the silent mode by current injection (Fig. 6). After 60 s of bursting activity, a sustained hyperpolarizing pulse current of -1 nA (bar) was injected into the cell to drive the membrane potential below the threshold of action potentials. Once bursting terminated, [cAMP] began to increase and stabilize at a level approximately 30% greater than during bursting. In the IE/B model, this increase was attributed to different responses of AC and PDE to a decrease in [Ca²⁺] (see DISCUSSION). The amount of increase was very similar to what was observed by Kramer et al. (1988). Thus, the IE/B model was able to accurately simulate an empirical observation that previous models could not.

View larger version (12K): [in this window] [in a new window]   FIG. 6. Level of cAMP is lower during bursting than during the silent state. Initially, a stable bursting mode was simulated. Silent state was induced by injecting a sustained -1 nA hyperpolarizing current into the cell (bar). In the silent state, [cAMP] increased by about 30%.

The anomalous rectifier current (I_R), which is active mostly during the interburst hyperpolarization, can be inactivated by Ca²⁺ influx (Kramer and Levitan 1988, 1990; Kramer et al. 1988; Fig. 7A). The phenomenon is termed activity-dependent inactivation of I_R because the Ca²⁺ influx was introduced by stimulating the neuron to generate bursting-like activity. Kramer et al. (1988, 1990) hypothesized that I_R was inactivated by Ca²⁺ influx in an indirect manner. Ca²⁺ did not act directly on I_R; instead, it was the cAMP-dependent activation of I_R that was inhibited by the Ca²⁺ influx. Application of 5-HT, which led to an elevation in the cAMP level, eliminated most of the inactivation of I_R caused by Ca²⁺ influx even while the magnitude of I_R was increased.

View larger version (17K): [in this window] [in a new window]   FIG. 7. Activity-dependent neuromodulation of anomalous rectifier (IR). A: experimental results demonstrating Ca2+-dependent inactivation of IR and inhibition of this effect by 5-HT (modified from Kramer et al. 1988). Each recording shows 2 superimposed hyperpolarizing voltage-clamp responses during a step from -80 to -110 mV. These inward currents were primarily attributed to IR (Kramer et al. 1988). Larger current was measured before Ca2+ influx and the smaller current (labeled with a dot) was measured after Ca2+ influx. Ca2+ influx was induced by depolarizing the cell to generate a "simulated burst" of spikes. Difference between the 2 superimposed currents corresponded to the portion of IR inactivated by Ca2+ influx. After application of 5-HT, inactivation of IR by Ca2+ influx diminished, although the amplitude of IR was increased. B: simulation following a similar procedure outlined in the experiment. Note that IR was directly monitored here instead of inferring IR from the inward current (see text).

The IE/B model faithfully reproduced the empirical data (Fig. 7B). Without 5-HT application, the simulation showed 20% activity-dependent inactivation of I_R, whereas the empirical result showed 28%. With 5-HT application, activity-dependent inactivation of I_R was reduced to 2%, whereas the empirical result showed 8% activity-dependent inactivation of I_R. The percentage changes in the simulations agreed well with the empirical results. Thus, these simulations suggest that activity-dependent inactivation of I_R can be quantitatively explained by inhibition of cAMP-dependent modulation of this current.

However, there was some discrepancy between our simulation and the experiment in absolute values. The I_R measured in the experiment was 20–40 nA in magnitude but the I_R measured in the simulation was only 10–15 nA in magnitude. This discrepancy is attributed, at least in part, to the fact that different currents were measured in the experiment (i.e., the total inward current of which I_R was only one component) and in the simulation (I_R only). For comparison, we also monitored the total inward current in the simulation. Results showed that before 5-HT application, I_R represented about 70–80% of the total inward current (the remaining 20–30% was mostly Na⁺–Ca²⁺ exchanger current). After 5-HT application, the portion of I_R in the total inward current dropped to about 50%. After taking this difference into consideration, the simulation result was not substantially different from the empirical data.

Role of oscillations of cAMP in bistability

Bistability, or multistability, is a dynamic property found in previous modeling as well as empirical studies of R15 (Bertram 1994; Butera et al. 1995; Canavier et al. 1991, 1993; Lechner et al. 1996). Specifically, bistability or multistability means that for a given set of parameters or experimental conditions, a neuron can express 2 or more different stable modes of activity (such as bursting or beating) depending on the initial value of variables or the history of activity within the system. Switching between the different modes can be achieved by applying appropriate perturbations.

Figure 8A1 illustrates an example of bistability found in the IE/B model. The model initially exhibited stable bursting activity until a brief hyperpolarizing current pulse (400 ms, -1 nA) was injected, which switched the model to the stable beating mode. Then the bursting mode was reinstated by another brief perturbation (5 s, -0.2 nA). In the parameter space, this bistability was found with a 5-HT concentration of 10 µM and a bias depolarizing current of 1.15 nA. However, bistability for this set of parameters (10 µM 5-HT, 1.15 nA bias current) was present only when [cAMP] was allowed to oscillate. After [cAMP] was clamped at a constant level, 2.18 µM, bistability disappeared (Fig. 8B1; 2.18 µM was the average [cAMP] when 10 µM 5-HT was applied).

View larger version (45K): [in this window] [in a new window]   FIG. 8. Importance of cAMP oscillations to bistability. A1: bistability displayed by the IE/B model. Initially, the model exhibited stable bursting activity (red trace). After a hyperpolarizing current pulse (-1 nA, 400 ms; the first arrowhead) was injected, the model switched to a stable beating mode (blue trace). The bursting mode was reinstated by another perturbation (-0.2 nA, 5 s; the second arrow). A2: phase-plane plot of activation of slow inward current (s) vs. internal free calcium concentration ([Ca]i). There were 2 attractors in the phase space: the bursting attractor (red trace) and the beating attractor (blue trace). B1: inhibition of bistability by clamping the level of cAMP. When [cAMP] was clamped at 2.18 µM, which was the average [cAMP] found in A, the same perturbation as in A failed to induce a transition from bursting mode to beating mode. B2: phase-plane plot of s vs. [Ca]i when [cAMP] was clamped. The bursting attractor was in the same position as in A2 but the beating attractor was not observed. Only several beating cycles were generated (green trace) after the perturbation and the trajectory returned to the bursting attractor.

Examination of phase-plane projections reveals the presence of two "attractors" (Fig. 8A2): one is associated with bursting (red trace) and the other associated with beating (blue trace). An appropriate perturbation (400 ms, -1 nA), when applied in the middle of a burst, can cause the trajectory to jump from the bursting attractor to the beating attractor. When [cAMP] was clamped, the bursting attractor appeared at the same position (Fig. 8B2, red trace). The same perturbation as described above caused the trajectory to deviate from the bursting attractor in a manner similar to A2. However, after several cycles of beating activity (green trace) in the region where the beating attractor was found before, the trajectory returned to the bursting attractor, indicating lack of a beating attractor in that region. To confirm this result, perturbation was varied in time (±1 s in 100-ms steps) and strength (-0.5 to -2.0 nA) but still no bistability was found.

Additional simulations found that bistability did not disappear completely when [cAMP] was clamped. Bistability could exist when the bias current was set within a specific range. However, the region of bistability was substantially expanded when there were oscillations in [cAMP] (Fig. 9). The range of bias current that allowed for bistability to occur was much larger when there were oscillations in [cAMP] than when [cAMP] was clamped at the average level. For example, when [5-HT] = 10 µM, this range was 1.00 to 1.29 nA when there were oscillations in [cAMP] and only 1.20 to 1.29 nA when [cAMP] was clamped at 2.18 µM. Compared with the model of Butera et al., bistability was found at all 5-HT concentrations examined from 0 to 100 µM, instead of being found only in a small range of 5-HT concentration. Across this large range of 5-HT concentrations, there appeared to be a continuous region of bistability between the region of the bursting mode and the region of the beating mode. Therefore oscillations in [cAMP] make bistability more robust and significantly expand the region of bistability.

View larger version (23K): [in this window] [in a new window]   FIG. 9. Region of bistability depends on oscillations of cAMP. Bistability is determined by 2 parameters: the bias current and [5-HT]. Between the upper and lower lines (the whole gray area) is the region of bistability when there are oscillations of cAMP. Between the upper line and the middle line (the hatched area) is the region of bistability when [cAMP] is clamped at its average level. Above the region of bistability there is only beating activity and below the region of bistability there is only bursting activity. To determine whether bistability existed at a certain point, phase-plane analysis as in Fig. 8 was used to explore the parameter space. Perturbation in various strengths (0.5–2.0 nA) and at different timing (in 100-ms steps) was applied to induce bistability.

	DISCUSSION

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Oscillations of cAMP

Because regulation of cAMP synthesis/degradation were linked with rhythmic electrical activity, it is not surprising that [cAMP] oscillated during bursting. The phase of these oscillations were somewhat counterintuitive. Changes in [cAMP] were determined by rates of synthesis and degradation, which correspond in our model to the activities of AC and PDE, respectively. According to the dose–response curves of AC and PDE (Fig. 2), PDE was more sensitive than AC to changes of [Ca²⁺] during the bursting activity. As the result, when the level of Ca²⁺ was high during a burst of spikes, degradation of cAMP outpaced synthesis and reduced the level of cAMP. Therefore, [cAMP] oscillated antiphasic to oscillations of [Ca²⁺] and oscillations of AC/PDE activity. The amplitude of cAMP oscillations was determined by the difference between the activities of AC and PDE activated by Ca²⁺/CaM. Greater differences led to larger-amplitude oscillations in [cAMP].

Although [cAMP] oscillated during bursting, examination of baseline activity indicated that these oscillations had very little effect on bursting activity. This was partly because the amplitude of cAMP oscillations was relatively small and partly because the 2 currents that cAMP upregulated, I_R and I_SI, have opposing actions during bursting and therefore canceled out part of each other's effect on the I–V curve. In the IE/B model, I_SI was still the key current responsible for inducing bursting (Butera et al. 1995; Canavier et al. 1991). Bursting starts with fast voltage activation of I_SI, continues with accumulation of internal Ca²⁺ and ends with slow inactivation of I_SI.

The present model was not the first model to predict oscillations of intracellular cAMP. Two previous models based on Ca²⁺–cAMP interaction also predicted that [cAMP] should oscillate. One of them predicted that the cAMP oscillations should be antiphasic to Ca²⁺ oscillations (Cooper et al. 1995), whereas the other predicted that the cAMP and Ca²⁺ oscillations should be in phase (Gorbunova and Spitzer 2002). The reason for such discrepancy was ascribed to different schemes of these 2 models. The model of Cooper et al. was based on a mammalian system and simulated only the downregulation of one type of AC (type V, found in the mammalian CNS) by free Ca²⁺, whereas the model of Gorbunova and Spitzer was based on cultured Xenopus cells and simulated only the upregulation of AC by Ca²⁺/CaM. Our model simulated both the upregulation of AC by Ca²⁺/CaM and the upregulation of PDE by Ca²⁺/CaM. The model of Cooper et al. relied on the negative feedback loop of cAMP synthesis to generate oscillations, whereas cAMP oscillations in the model of Gorbunova and Spitzer and our model were driven by the spontaneous Ca²⁺ transients or the bursting activity. It is interesting to note that the model of Cooper et al. (1995), which was based primarily on biochemical mechanisms, predicted a cycle period of about 10 s, which coincided with the cycle period of R15 (10–20 s). If R15 had an intrinsic oscillator based on its biochemical properties (Ca²⁺/CaM–cAMP interactions), it is possible that this "biochemical oscillator" by itself would have a cycle period of approximately 10 s. If this is the case, the "biochemical oscillator" in R15 is well coupled with the "electrical oscillator."

Evidence for cAMP oscillations have been found in several previous studies. Oscillations in [cAMP] were detected during rhythmic myocardial contraction of the frog ventricle (Brooker 1973). The average period of these oscillations was only 2 s. A slime mold, Dictyostelium discoideum, displays waves of cAMP during periodic aggregation (Tomchik and Devreotes 1981). Moreover, spontaneous intracellular oscillations of cAMP were recorded in cell suspensions of Dictyostelium discoideum (Gerisch and Wick 1975; Gerisch et al. 1975). These oscillations are believed to be important for cell communication and synchronizing movements among cells. More recently, oscillations of cAMP have been suggested to occur in frog olfactory receptor cells (Reisert and Matthews 2001). The study by Gorbunova and Spitzer (2002) on Xenopus culture neurons also showed that specific patterns of oscillating Ca²⁺ transients drove the cAMP levels to oscillate with substantial amplitude.

Oscillations of cAMP levels may have significant biological implications. As an important second messenger, cAMP acts on many enzymes, receptors, and channels. Because of the nonlinear properties of the second-messenger cascades, oscillating [cAMP] may exert substantially different effects on downstream targets than would a constant level of cAMP. For example, pulsatile cAMP changes are more efficient in inducing the release of certain hormones (Haisenleder et al. 1992; Vitalis et al. 2000). Because R15 releases bioactive peptides important for cardiovascular, digestive, respiratory, and reproductive systems (Alevizos et al. 1991), it is possible that oscillating cAMP would facilitate such transmitter release. Moreover, simultaneous and interdependent Ca²⁺ and cAMP oscillations may generate distinct intracellular signaling patterns that are required for activation of certain kinases or transcriptional regulation (Zaccolo and Pozzan 2003).

In recent years, new techniques for detecting intracellular cAMP emerged by use of a protein kinase A–derived fluorosensor, FlCRhR (Adams et al. 1991, 1993; Gorbunova and Spitzer 2002) or by use of genetically modified cyclic nucleotide-gated channels as cAMP sensors (Rich et al. 2000, 2001). The sensitivity of these methods should be adequate for detecting the level of cAMP expected in Aplysia neurons. If one of these methods can detect a concentration change as small as 10% predicted by this model, it would be possible to empirically examine the oscillations of cAMP in R15.

Activity-dependent inactivation of anomalous rectifier

Activity-dependent inactivation of I_R can also be explained by the difference in the sensitivity of AC and PDE to Ca²⁺/CaM. Because PDE was more sensitive to Ca²⁺/CaM than AC, more cAMP was degraded than synthesized when the Ca²⁺ level was elevated. Therefore the net effect of Ca²⁺ influx was to lower the level of cAMP. Because cAMP upregulates I_R (Fig. 1) and Ca²⁺ influx lowers the level of cAMP, Ca²⁺ influx indirectly inactivated I_R via changes in the level of cAMP. When 5-HT was applied, the cAMP level was elevated and overcame the response to Ca²⁺ influx and therefore Ca²⁺-dependent inactivation of I_R was diminished.

The time course of activity-dependent inactivation of I_R was different in the simulation as compared to the empirical results. In experiments, inactivation of I_R took about 90 s to reach its peak (Kramer and Levitan 1990; Kramer et al. 1988). The delay was presumably attributable to the time required for cAMP to modulate the conductance of I_R, through phosphorylation of channel proteins by a protein kinase. There was no such delay in the simulation and therefore inactivation of I_R peaks immediately after the Ca²⁺ influx. This discrepancy may be corrected if additional kinetic steps or appropriate time delays are added to the simulation.

Different cAMP levels during the bursting and the resting state

The difference in the sensitivity of AC and PDE to Ca²⁺/CaM was again the cause for the different cAMP levels during the bursting and the resting states. When the cell was hyperpolarized, the Ca²⁺ level drops to about 0.1 µM. According to the dose–response curves in Fig. 2, both AC activity and PDE activity decreased. However, the degree of decrease in activity was different for AC and PDE. The PDE activity decreased more than the AC activity because the PDE activity had a steeper dose–response curve between 0.1 and 1 µM. As a result, cAMP synthesis outpaced degradation and [cAMP] increased once the cell was hyperpolarized. Because the enzyme activity of PDE was [cAMP] dependent and it increased when [cAMP] became high, balance between synthesis and degradation was eventually achieved at a cAMP level that was higher than during the bursting state.

Bistability in R15

Bistability or multistability has been reported in many types of neurons including the lobster stretch receptor (Calvin and Hartline 1977), rat cortical neurons (Egorov et al. 2002), mammal spinal motoneurons (Lee and Heckman 1998), and rat olfactory bulb mitral cells (Heyward et al. 2001). Bistability in R15 was also found experimentally (Lechner 1996) after predictions by Canavier et al. (1993) and Butera et al. (1995) in their mathematical models. Although the functional significance of coexisting bursting and beating modes in R15 is still unknown, several possibilities have been discussed (Canavier et al. 1994; Lechner et al. 1996; Marder et al. 1996). Different modes of activity in R15 may have different effectiveness in transmitter/hormone release or different responsiveness to sensory inputs, and stable transitions among different modes of activities may serve as a type of memory.

The results in the present study suggest that the range of bistability becomes larger when interactions with second-messenger cascades are introduced into the system. This finding eases the concern whether bursting neurons ever enter the region of parameter space that supports bistability or multistability (Canavier et al. 1994). Considering that our current model is still a simplification of the complicated interactions within the cell, further extensions may prove that bistability or multistability is even more robust and widespread than we found in our current model. Because bistability or multistability has been observed in many models that produce bursting activity (Bertram 1994; Butera 1998; Destexhe et al. 1993; Guckenheimer and Holmes 1983; Rinzel and Lee 1987), the result of the present study should have implications to these models.

In conclusion, the IE/B model predicted that oscillations of the cAMP levels may result from interactions between the electrical activity and the biochemical cascades. Such oscillations are attributed to different responses of AC and PDE to oscillations of the Ca²⁺ levels. This prediction may be tested by monitoring the cAMP levels during bursting (Adams et al. 1991, 1993; Rich et al. 2000, 2001). Moreover, this study suggests that interactions between the electrical activity and the biochemical cascades enrich the dynamic complexity of neurons and new insights can be gained into the neuronal functions by modeling these interactions.

	APPENDIX: EQUATIONS AND PARAMETERS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

 
(Units: ms for time; nA for current; mV for potential; mM for concentration)

Inward currents

I_Na: FAST SODIUM CURRENT

I_Ca: FAST CALCIUM CURRENT

I_SI: SLOW INWARD CURRENT

I_NS: NONSPECIFIC CATION CURRENT

I_L: LEAKAGE CURRENT

Outward currents

I_K: DELAYED RECTIFIER

I_R: ANOMALOUS RECTIFIER

Pumps and exchangers

Internal calcium concentration

Internal cAMP concentration

	ACKNOWLEDGMENTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

GRANTS

This research was supported by National Institute of Neurological Disorders and Stroke Grant P01 NS-38310.

	FOOTNOTES

 
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Address for reprint requests and other correspondence: D. A. Baxter, Department of Neurobiology, The University of Texas–Houston Medical School, Houston, TX 77030 (E-mail: Douglas.Baxter{at}uth.tmc.edu).

	REFERENCES

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX: EQUATIONS AND... ACKNOWLEDGMENTS REFERENCES

 
Abrams TW, Karl KA, and Kandel ER. Biochemical studies of stimulus convergence during clas